This problem will help illustrate the terms defined above as well as the basic logic for all of hypothesis testing.
Letís say you use a coin to help you make decisions. 'Heads' means you study and 'tails' means you socialize. However, you feel that you are getting the raw end of the deal and believe the coin is biased. Thus, the research question asks whether the coin is biased. Let us examine the method for answering this question. Before we are ready to start flipping the coin a bunch of times and test the hypothesis, we need to get a deeper understanding of the concept of a Binomial Distribution that we defined above.

Constructing the Binomial Distribution

Explanation

First let us consider what can happen if we toss a coin once (i.e., N=1). The possible outcomes are:

A head (H) or a tail (T).

In summary, the binomial distribution for N=1 is:

Possible outcomes

H1T0

H0T1

Probability

1/2

1/2

Next let us consider what can happen if we toss a coin twice (i.e., N=2). The possible outcomes are:

2 heads (H2), a head and a tail (HT), and 2 tails (T2).

Note, however, that the head and a tail can occur in 2 ways (HT or TH). Thus, the possible outcomes are:

2 heads (1H2), a head and a tail (2H1T1),
and 2 tails (1T2).

The bold numbers are called coefficients and indicate the number
of different ways an outcome can occur. Note, that the sum of the coefficients
is equal to all possible outcomes (i.e., 1 + 2 + 1 = 4).

In summary, the binomial distribution for N=2 is:

Possible outcomes

H2T0

2H1T1

H0T2

All possible outcomes

HH

HTTH

TT

Probability

1/4

2/4

1/4

Notes:

As with all sampling distributions, all possible values of a statistic and their probabilities of occurrence are given.

N+1 = the number of different possible outcomes. In the present case, there are 3 different possible outcomes (i.e., H2T0, H1T1, H0T2).

2N = ∑coefficients = the total
number of all possible outcomes. In the present case, there
are 4 possible outcomes (i.e., HH, HT, TH, TT).

The probabilities for all outcomes sum to 1.

Now consider tossing the coin three times (i.e., N=3):

Different Possible outcomes

H3T0

H2T1

H1T2

H0T3

∑coefficients=8

All
possible outcomes

HHH

HHT
HTHTHH

HTT
THT
TTH

TTT

Coefficients
(or # all
possible outcomes)

1

3

3

1

Thus, the binomial distribution for N=3 is:

Possible outcomes

H3T0

H2T1

H1T2

H0T3

Probability

1/8

3/8

3/8

1/8

As you might guess, constructing a binomial when N>3 gets to be difficult
and thus there are more efficient methods.

Note that the exponent equals the number of tosses and the formula gives the coefficients. However, this, too,
is a tedious technique.

Pascalís Triangle

A trick with numbers that generates the coefficients.

N

Coefficients

N+1

2n

1

1 1

2

2

2

1 2 1

3

4

3

1 3 3 1

4

8

4

1 4 6 4 1

5

16

5

1 5 10 10 5 1

6

32

6

1 6 15 20 15 6 1

7

64

7

1 7 21 35 35 21 7 1

8

128

8

1 8 28 56 70 56 28 8 1

9

256

9

1 9 36 84 126 126 84 36 9 1

10

512

10

1 10 45 120 210 252 210 120 45 10 1

11

1024

To build the pyramid, start with 1's on the outside. For the inner numbers of the pyramid, add the above two numbers, that is:

Thus, this technique works well with small N's.

Testing HypothesesThis is basically a more in depth view of significance of differences that we discussed breifly at the beginning of the semester.
Back to our question. Is the coin funny?

State the Hypotheses
If we let p equal the probability of a head, and q equal the probability of a tail, then we can make two mutually exclusive statements or hypotheses
as follows. Note, mutually exclusive means that only one of the
statements can be true.

Hypothesis Name

Meaning

In Symbols

Comments

Null or Ho

The coin is fair

p=q

Always an exact statement

Alternative or HA or H1

The coin is funny

p≠q

Never an exact statement

Choose a Significance Level
The alpha level (α) is the arbitrary
level of significance that statisticians have chosen to distinguish probable
from improbable. The alpha level chosen in Psychology is typically .05,
with .01 or even .001 used in some circumstances.

Improbable
Due to Chance

Probable
Due to Chance

Assume the Null Hypothesis
We do this because the
null is an exact statement and therefore testable. In other words, it allows us to compute the relevant probabilities. In the case of the
coin, we assume it is fair and compute the probability that the outcome
we observed is due to chance. If the probability
of this event occurring due to chance is small (i.e., less than or equal
to the alpha level), we will reject the null hypothesis and assert
the alternative (that the coin is funny).

Describe & Compute the Probability of the Observed Outcome Describe the data (what is the outcome?) using the techniques and concepts you learned during the first part of the semester. Then we need to
compute the probability of an event "as
rareas" what we observed. Note that we do not simply compute the probability of the specific event we observed, rather we compute the probability of an event as
rare as what we observed. This is an essential part of the logic of hypthesis testing. When computing this probability, we use an inferential test. Which test we use depends on several factors. This table summarizes a number of tests and can help you decide which test is appropriate for a given data set. During the remaining portion of the semester, we will learn about a number of these tests.

Make a Decision
Compare the probability of the observed outcome to the alpha level and make a decision. Look closely at what this decision means for the data involved.

Other Relevant Issues.

Directional vs. Non-directional Hypotheses

Alternative Hyp.

Type

Meaning

Test to be used

p≠q

Non-directional

Coin is biased.

Two-tailed

p<q

Directional

Coin is biased for T.

One-tailed

p>q

Directional

Coin is biased for H.

One-tailed

Nondirectional tests are most common. Directional hypotheses are sometimes
used when we have a theory and/or prior data that leads us to such a
specific prediction. To keep things simple, in this class, we will use
two-tailed tests exclusively.

Accepting, Asserting, and Rejecting
We never "accept" the null or alternative hypotheses.
We either:

Reject the null and "assert" the alternative
We donít accept the alternative because we didnít test it. We tested
the null by assuming its truth.

"Fail to reject" the null.
We donít accept the null because maybe our test wasnít sensitive enough
to detect a bias in the coin.

Probabilistic Nature of Science
Given an alpha level of .05, we can expect to reject the null 1 in 20
times when the coin is actually fair. In other words, a fair coin could
give 10 heads in 10 tosses, it is just not very probable. Fortunately
though, science progresses when the study is replicated (and extended)
by the same and other investigators. The probability of making two mistakes
would be .05 x .05 = .0025. We will talk more about these "mistakes"
in the next chapter. An important moral of this story is that we never
used the word "prove" when talking about statistical
results.

Formal Example - [Minitab]
Let us look at an example that will set the stage for the format we will use when testing a hypothesis. We tossed the coin 10 times and got 9 heads.

Research Question
Is the coin biased?

Hypotheses

Let p equal the probability of a head and q the probability of a tail.

In Symbols

In Words

Ho

p=q

The coin is fair

HA

p≠q

The coin is funny

These hypotheses should be specific to what is being tested (in this case, the coin).

Assumptions

The null hypothesis (i.e., Ho).

Decision Rules
We will use an Alpha of .05 with a two-tailed test.
If the probability of what we observe ≤ .05, reject Ho.
If the probability of what we observe > .05, fail to reject Ho.

Computation
The computations have two goals corresponding to the descriptive and inferential
statistics.

The first step is to describe the data. In this case, we tossed the coin 10 times and got 9 heads. That is, 90% when we would expect 50%. Thus, the appropriate descriptive statistic in this case is a percent.

The second
is to perform an inferential test which in this case, is a binomial test because we are dealing with a dichotomous variable. Thus, so we need Pascal’s triangle for N=10 in order to obtain the relevant sampling distribution. Since we already created one of those above, we can just take the relevant information from there. What follows is the binomial
distribution for N=10.

Possible
outcomes

H10T0

H9T1

H8T2

H7T3

H6T4

H5T5

H4T6

H3T7

H2T8

H1T9

H0T10

P

1/1024

10

45

120

210

252

210

120

45

10

1

.001

.010

.044

.117

.205

.246

.205

.117

.044

.010

.001

The probabilities of an event as rare as 9 heads are shown in color above and are summed
below.

Event

Probability

10 heads

.001

9 heads

.010

9 tails

.010

10 tails

.001

Total

= .022 (or .011 x 2)

Decision
The probability of observing an event as rare as 9 heads in 10 tosses of a fair coin is .022. Since
this is less than the alpha level of .05, we reject Ho and assert the alternative. In other words, we conclude that the
outcome we have observed is improbable due to chance and the coin is biased. Furthermore, we can state that it is biased in the heads direction.

Notice that we have rejected the null and asserted the alternative which stated that the coin is funny. Then we explained how it is funny. Thus, in this section, we need to make a decision about rejecting or failing to reject the null, and, we must tell what our decision means in terms of the variables involved.

If our calculations had given a probability of .051 instead of .022, we would have failed to reject the null and concluded that the coin is fair.